2026-07-072026-07-07http://salesiana.dossiersoluciones.com/handle/123456789/163883We prove nonexistence of nonconstant local minimizers for a class of functionals, which typically appears in the scalar two-phase field model, over a smooth N-dimensional Riemannian manifold without boundary with non-negative Ricci curvature. Conversely for a class of surfaces possessing a simple closed geodesic along which the Gauss curvature is negative we prove existence of nonconstant local minimizers for the same class of functionals.26 pagesDifferential GeometryFunctional Analysis35J20; 58J05text